3.1518 \(\int \frac{(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \]

[Out]

-(d + e*x)^5/(5*(b*d - a*e)*(a + b*x)^5)

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Rubi [A]  time = 0.0233853, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{(d+e x)^5}{5 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(d + e*x)^5/(5*(b*d - a*e)*(a + b*x)^5)

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Rubi in Sympy [A]  time = 12.5426, size = 20, normalized size = 0.71 \[ \frac{\left (d + e x\right )^{5}}{5 \left (a + b x\right )^{5} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(d + e*x)**5/(5*(a + b*x)**5*(a*e - b*d))

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Mathematica [B]  time = 0.100273, size = 140, normalized size = 5. \[ -\frac{a^4 e^4+a^3 b e^3 (d+5 e x)+a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a b^3 e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{5 b^5 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a
*b^3*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10
*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))/(5*b^5*(a + b*x)^5)

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Maple [B]  time = 0.01, size = 185, normalized size = 6.6 \[ 2\,{\frac{{e}^{3} \left ( ae-bd \right ) }{{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{e \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{{e}^{4}{a}^{4}-4\,d{e}^{3}{a}^{3}b+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}-{\frac{{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}-2\,{\frac{{e}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{b}^{5} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^3*(a*e-b*d)/b^5/(b*x+a)^2+e*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/b^
5/(b*x+a)^4-1/5*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/
b^5/(b*x+a)^5-e^4/b^5/(b*x+a)-2*e^2*(a^2*e^2-2*a*b*d*e+b^2*d^2)/b^5/(b*x+a)^3

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Maxima [A]  time = 0.693804, size = 290, normalized size = 10.36 \[ -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^
4*e^4 + 10*(b^4*d*e^3 + a*b^3*e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2
*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a^3*b*e^4)*x)/(b^10*x
^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Fricas [A]  time = 0.197518, size = 290, normalized size = 10.36 \[ -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

-1/5*(5*b^4*e^4*x^4 + b^4*d^4 + a*b^3*d^3*e + a^2*b^2*d^2*e^2 + a^3*b*d*e^3 + a^
4*e^4 + 10*(b^4*d*e^3 + a*b^3*e^4)*x^3 + 10*(b^4*d^2*e^2 + a*b^3*d*e^3 + a^2*b^2
*e^4)*x^2 + 5*(b^4*d^3*e + a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + a^3*b*e^4)*x)/(b^10*x
^5 + 5*a*b^9*x^4 + 10*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Sympy [A]  time = 16.7861, size = 233, normalized size = 8.32 \[ - \frac{a^{4} e^{4} + a^{3} b d e^{3} + a^{2} b^{2} d^{2} e^{2} + a b^{3} d^{3} e + b^{4} d^{4} + 5 b^{4} e^{4} x^{4} + x^{3} \left (10 a b^{3} e^{4} + 10 b^{4} d e^{3}\right ) + x^{2} \left (10 a^{2} b^{2} e^{4} + 10 a b^{3} d e^{3} + 10 b^{4} d^{2} e^{2}\right ) + x \left (5 a^{3} b e^{4} + 5 a^{2} b^{2} d e^{3} + 5 a b^{3} d^{2} e^{2} + 5 b^{4} d^{3} e\right )}{5 a^{5} b^{5} + 25 a^{4} b^{6} x + 50 a^{3} b^{7} x^{2} + 50 a^{2} b^{8} x^{3} + 25 a b^{9} x^{4} + 5 b^{10} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(a**4*e**4 + a**3*b*d*e**3 + a**2*b**2*d**2*e**2 + a*b**3*d**3*e + b**4*d**4 +
5*b**4*e**4*x**4 + x**3*(10*a*b**3*e**4 + 10*b**4*d*e**3) + x**2*(10*a**2*b**2*e
**4 + 10*a*b**3*d*e**3 + 10*b**4*d**2*e**2) + x*(5*a**3*b*e**4 + 5*a**2*b**2*d*e
**3 + 5*a*b**3*d**2*e**2 + 5*b**4*d**3*e))/(5*a**5*b**5 + 25*a**4*b**6*x + 50*a*
*3*b**7*x**2 + 50*a**2*b**8*x**3 + 25*a*b**9*x**4 + 5*b**10*x**5)

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GIAC/XCAS [A]  time = 0.211908, size = 230, normalized size = 8.21 \[ -\frac{5 \, b^{4} x^{4} e^{4} + 10 \, b^{4} d x^{3} e^{3} + 10 \, b^{4} d^{2} x^{2} e^{2} + 5 \, b^{4} d^{3} x e + b^{4} d^{4} + 10 \, a b^{3} x^{3} e^{4} + 10 \, a b^{3} d x^{2} e^{3} + 5 \, a b^{3} d^{2} x e^{2} + a b^{3} d^{3} e + 10 \, a^{2} b^{2} x^{2} e^{4} + 5 \, a^{2} b^{2} d x e^{3} + a^{2} b^{2} d^{2} e^{2} + 5 \, a^{3} b x e^{4} + a^{3} b d e^{3} + a^{4} e^{4}}{5 \,{\left (b x + a\right )}^{5} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

-1/5*(5*b^4*x^4*e^4 + 10*b^4*d*x^3*e^3 + 10*b^4*d^2*x^2*e^2 + 5*b^4*d^3*x*e + b^
4*d^4 + 10*a*b^3*x^3*e^4 + 10*a*b^3*d*x^2*e^3 + 5*a*b^3*d^2*x*e^2 + a*b^3*d^3*e
+ 10*a^2*b^2*x^2*e^4 + 5*a^2*b^2*d*x*e^3 + a^2*b^2*d^2*e^2 + 5*a^3*b*x*e^4 + a^3
*b*d*e^3 + a^4*e^4)/((b*x + a)^5*b^5)